Axiomatic Reduction of Σ_min: Complete Proofs for the Companion Paper "Axiomatic Derivation of the Uniqueness of the Fano Plane from a Non-Freezing Principle"

This paper provides the detailed proofs cited as available in [ReductionAxiomatique] in the companion paper (Zenodo, 2026, DOI: [ton DOI v10]). Three results are established: (I) Theorem A0: The unique minimal host structure compatible with Σ_min = {A1, A2, Causality, MP(III)} is Z⁸. The elimination of d ∈ {1, 2, 4} is complete and non-circular. (II) Theorem A4: G₂ = Aut(O) is proved in 5 explicit steps from the Fano structure: construction of 14 generators as the kernel of the Fano 3-form action on so(7), verification of dimension 14, semi-simplicity via the Killing form, simplicity via irreducibility of the action on R⁷, and identification via the Cartan-Killing classification. A6 (rank 2) and A8 (condensate e₃ → SU(3)) follow as corollaries. (III) Formal construction of σ₃ and evolve from Σ_min, including independence from the observation radius r_obs and the canonical tie-breaking rule. (IV) SQ-R P-admissibility transfer: complete verification that the replacement construction C(B+) preserves P1–P5 and N1–N4, with explicit construction guaranteeing P3 for each block. The only numerical result not reproduced inline is det(B_K) = 4.03 × 10¹⁹ (Step 3 of Theorem A4), which is computed explicitly from the 14 generators in Doc 306 of the Triniton corpus (available upon request). The Cartan-Killing classification is used as a standard auxiliary theorem. Zero free parameters. Zero ad hoc adjustments.